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1、定义
载体坐标系b到导航坐标系n的坐标转换矩阵CbnC_b^nCbn​ 或 TTT。

2、欧拉角求解
从导航坐标系依次转过航向角 ψ\psiψ 、俯仰角 $ \theta$ 和滚转角 $ \gamma $ 可得载体坐标系..." />
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1、定义
载体坐标系b到导航坐标系n的坐标转换矩阵CbnC_b^nCbn​ 或 TTT。

2、欧拉角求解
从导航坐标系依次转过航向角 ψ\psiψ 、俯仰角 $ \theta$ 和滚转角 $ \gamma $ 可得载体坐标系...">
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1、定义
载体坐标系b到导航坐标系n的坐标转换矩阵CbnC_b^nCbn​ 或 TTT。

2、欧拉角求解
从导航坐标系依次转过航向角 ψ\psiψ 、俯仰角 $ \theta$ 和滚转角 $ \gamma $ 可得载体坐标系...">
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                                <h2>
                                    5.3 捷联矩阵
                                </h2>
                                <span class="article-info">
                                    2023-12-17, 589 words, 3 min read
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                                        <h2 id="一-捷联矩阵">一、捷联矩阵</h2>
<h4 id="1-定义">1、定义</h4>
<p>载体坐标系b到导航坐标系n的坐标转换矩阵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>C</mi><mi>b</mi><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">C_b^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9664379999999999em;vertical-align:-0.2831079999999999em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4168920000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span>。</p>
<img src="http://cos.pansis.site/202312141431260.png/abc123" alt="image-20231214143104330" style="zoom:33%;" />
<h4 id="2-欧拉角求解">2、欧拉角求解</h4>
<p>从导航坐标系依次转过航向角 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">ψ</span></span></span></span> 、俯仰角 $ \theta$ 和滚转角 $ \gamma $ 可得载体坐标系</p>
<img src="http://cos.pansis.site/202312141433531.png/abc123" alt="image-20231214143330477" style="zoom:50%;" />
<img src="http://cos.pansis.site/202312141433037.png/abc123" alt="image-20231214143358960" style="zoom:33%;" />
<img src="http://cos.pansis.site/202312141434047.png/abc123" alt="image-20231214143409982" style="zoom:50%;" />
<h4 id="3-捷联矩阵即时修正的算法">3、捷联矩阵即时修正的算法</h4>
<ul>
<li>欧拉角法（三参数法）</li>
<li>四元数法（四参数法）</li>
<li>方向余弦法（九参数法）</li>
</ul>
<h2 id="二-欧拉角法">二、欧拉角法</h2>
<h4 id="1-算法">1、算法</h4>
<p>载体坐标系相对导航坐标系的姿态角速度可表示为三次欧拉转动角速度的矢量和</p>
<img src="http://cos.pansis.site/202312141437540.png/abc123" alt="image-20231214143720496" style="zoom:50%;" />
<p>其中，<img src="http://cos.pansis.site/202312141437708.png/abc123" alt="image-20231214143742652" style="zoom:33%;" /></p>
<img src="http://cos.pansis.site/202312141438402.png/abc123" alt="image-20231214143848338" style="zoom:53%;" />
<p>求逆得</p>
<img src="http://cos.pansis.site/202312141439299.png/abc123" alt="image-20231214143933234" style="zoom:50%;" />
<h4 id="2-局限性">2、局限性</h4>
<ul>
<li>求解欧拉角微分方程只需要解三个微 分方程。与其它的算法相比，要解的方程数少些但在用计 算机进行数值积分时要进行超越函数的运算，这反而加大了计算的工作量。</li>
<li>当<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi><mo>=</mo><mn>90</mn><mi mathvariant="normal">°</mi></mrow><annotation encoding="application/x-tex">\theta=90°</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mord">0</span><span class="mord">°</span></span></span></span>时将出现奇点，无法计算</li>
</ul>
<h2 id="三-四元数法">三、四元数法</h2>
<h4 id="1-四元数的性质">1、四元数的性质</h4>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Q</mi><mo>=</mo><msub><mi>q</mi><mn>0</mn></msub><mo>+</mo><msub><mi>q</mi><mn>1</mn></msub><mi>i</mi><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><mi>j</mi><mo>+</mo><msub><mi>q</mi><mn>3</mn></msub><mi>k</mi></mrow><annotation encoding="application/x-tex">Q=q_{0}+q_{1}i+q_{2}j+q_{3}k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span></p>
<p>1、共轭：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>Q</mi><mo>∗</mo></msup><mo>=</mo><msub><mi>q</mi><mn>0</mn></msub><mo>−</mo><msub><mi>q</mi><mn>1</mn></msub><mi>i</mi><mo>−</mo><msub><mi>q</mi><mn>2</mn></msub><mi>j</mi><mo>−</mo><msub><mi>q</mi><mn>3</mn></msub><mi>k</mi></mrow><annotation encoding="application/x-tex">Q^*=q_{0}-q_{1}i-q_{2}j-q_{3}k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831359999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span></span></span></span></p>
<p>2、范数：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>N</mi><mo>=</mo><mi>Q</mi><msup><mi>Q</mi><mo>∗</mo></msup><mo>=</mo><msubsup><mi>q</mi><mn>0</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>q</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>q</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>q</mi><mn>3</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">N=QQ^*=q_{0}^2+q_{1}^2+q_{2}^2+q_{3}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8831359999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">Q</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0622159999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0622159999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0622159999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0622159999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span> 当N=1时，四元数 Q 为单位四元数。</p>
<p>3、逆：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>Q</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><msup><mi>Q</mi><mo>∗</mo></msup><mi>N</mi></mfrac></mrow><annotation encoding="application/x-tex">Q^{-1}=\frac{Q^*}{N}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.325448em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.980448em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7633428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
<h4 id="2-四元数描述刚体转动">2、四元数描述刚体转动</h4>
<p>1、坐标系转换</p>
<ul>
<li>b系由n系绕z轴转动30°得到，可得</li>
</ul>
<img src="http://cos.pansis.site/202312141447041.png/abc123" alt="image-20231214144722991" style="zoom:33%;" />
<p>2、坐标转换</p>
<p>矢量S在坐标系b和n中的坐标转换：<img src="http://cos.pansis.site/202312141449693.png/abc123" alt="image-20231214144913658" style="zoom:33%;" /></p>
<p>3、三次转动</p>
<p>从导航坐标系依次转过航向角 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">ψ</span></span></span></span> 、俯仰角 $ \theta$ 和滚转角 $ \gamma $ 可得载体坐标系</p>
<img src="http://cos.pansis.site/202312141449693.png/abc123" alt="image-20231214144913658" style="zoom:33%;" />
<p>其中，<img src="http://cos.pansis.site/202312141450998.png/abc123" alt="image-20231214145027959" style="zoom: 33%;" /><img src="http://cos.pansis.site/202312141451345.png/abc123" alt="image-20231214145102283" style="zoom:33%;" /></p>
<p>最终可得，<img src="http://cos.pansis.site/202312141451890.png/abc123" alt="image-20231214145126843" style="zoom:50%;" /></p>
<h4 id="3-修正算法">3、修正算法</h4>
<img src="http://cos.pansis.site/202312141452516.png/abc123" alt="image-20231214145239470" style="zoom:33%;" />
<h4 id="4-特点">4、特点</h4>
<p>在进行数值积分求解时只需要进行加减法与乘法运算，求解的计算 量要比欧拉角法少得多</p>
<h2 id="四-方向余弦法">四、方向余弦法</h2>
<h4 id="1-算法-2">1、算法</h4>
<p>方向余弦矩阵的微分方程为<img src="http://cos.pansis.site/202312141456846.png/abc123" alt="image-20231214145659801" style="zoom:30%;" /></p>
<h4 id="2-特点">2、特点</h4>
<ul>
<li>需要解9个微分方程，同样也只要进行加 减法和乘法运算，显然求解方向余弦矩阵微分方程要比四 元数微分方程的方程数多</li>
<li>然而采用方向余弦法可以直接求出捷联矩阵 T</li>
</ul>
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